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Good morning, I report here an equation that you can find in the following paper: "Portfolio Selection with Options and Transaction Costs" by Semyon Malamud (2014) page 10 - 11 (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2397760)

Suppose that the state price $\eta_{\left(i,\theta\right)}$ for a stock $i$ depends on both its own price and its observable volatility $v_{i.t}$ so that $\eta_{\left(i,\theta\right)}\left(X_t,y\right)=\frac{1}{\sigma_t}\left(\left(y - X_{i,t}\right)\right)/v_{i,t}$. Let $k_{X_i}$ and $k_{v_i}$ denote the components of $k$ corresponding to the derivatives with respect to $X_i$ and $v_i$. For any $k$ with only $k_{X_i}$ and $k_{v_i}$ non zero, I have:

\begin{equation} \eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(X_t,y\right)=\frac{\partial^{k_{X_i}+k_{v_i}}}{\partial X_i^{k_{X_i}}\partial v_i^{k_{v_i}}}\frac{1}{\sigma_t}\eta_{\left(i,\theta\right)}\left(\frac{\left(y - X_{i,t}\right)}{v_{i,t}}\right) =\sum_{k\leq |\textbf{k}|}^{} B_k\left(X_t\right)\eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(\frac{\left(y - \bar{X}_{i,t}\right)}{v_{i,t}}\right) \end{equation}

Consequently the span of $\eta_{\left(i,\theta\right)}^{\left(\textbf{k}\right)}\left(X_t,y\right)$, $k_{X_i}+k_{v_i}\leq k$ has dimension $k$. If claims on volatility $v_{i,t}$ are tradable and the corresponding conditional state prices are given by:

\begin{equation} \eta_{\left(v_i,\theta\right)}\left(X_t,y\right)=\eta_{\left(v_i,\theta\right)}\left(X_{i,t},v_{i,t},y\right) \end{equation} the span of the corresponding derivatives will be larger.

The equation and the subsequent comments are quite obscure to me. The coefficients $B_k\left(X_t\right)$ are not even defined throughout the paper. I can just think of some application of a Taylor expansion.

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